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On the (Small) Number of Atoms in the Universe

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Re: On the (Small) Number of Atoms in the Universe

#91

Post by ambulancechaser »

levemi wrote: post #85
Retric wrote: post #70

Earlier quoted context omitted.

Math includes the idea of orders of infinity. There are infinite prime numbers, there are more positive integers, even more integers (positive and negative), even more rational numbers (A/B), and even more numbers (rational + irrational {e, Pi} etc)...

Within a segment of numbers I understand how there are more rational numbers than integers, but I don't understand it in the context of infinity. How can there be more rational numbers than integers when in both cases there are infinite amounts? Are there mathematical operations or concepts that depend on this (in the context of infinity, not subsets)?

in mathematics, we lose concept of how many and fall back to cardinality, which has a lose correlation with how many. So asking "are there the same number" of integers as rational numbers gets a little iffy until we make some definitions. We just say that we can create a bijection from integers to rationals. They each index the other, and for each thing in one, there is one and only one thing in the other. Does this mean there are the same "many"? Well loosely, and in the context of cardinality, yes. But things get weird because there are the same "many" of the whole as a subset (ie, there are as "many" even numbers as integers, as "many" positive numbers as positive and negative integers, etc).

But most people would disagree with you when you say "how can there be more rational numbers than integers..." because while we don't have a firm grasp of how many, we definitely would say that having the same cardinality means that there isn't some notion of "more".

I'm not sure what you mean by mathematical operations or concepts that depend on this.

Re: On the (Small) Number of Atoms in the Universe

#92

Post by dack »

kagia wrote: post #79

If 12megapixels can produce 10 to the power 86696638 images, and we came up with a way of enumerating those images, could we then build a function that given anyone of those images return the index of that image within reasonable time with current hardware. ie. "you have just taken 3999999987493th image"?

not only could you enumerate all images, you can also enumerate all books. the library of babel does this and youre looking at the particular page where this exact paragraph is written, for hacker news by dackerman, no less.

https://libraryofbabel.info/bookmark.cgi?ci.pnvgmzgyrce29

Re: On the (Small) Number of Atoms in the Universe

#93

Post by Retric »

pcmonk wrote: post #86
Retric wrote: post #84

Earlier quoted context omitted.

The set of all prime numbers is contained within the set of rational numbers, but they are rational numbers that are not within the set of prime numbers. Cantor's diagonalization is simply demonstrating that same inequality by showing a number in set A is not in set B. Just because you can map two infinity's to each other does not mean they are of the same size consider: Limit(0->inifinity) of (x - (x/2)) algebraical…

Where in math is the subset partial ordering used to describe one set as larger than another? Diagonalization isn't showing that a number in set A isn't in set B - that's obviously true for reals and integers, but it's also true for rationals and integers. It's showing that there does not exist a mapping from B to A where there's an element in B for each element in A. We're obviously not using the same definition of…

Cardinality is the number of elements in a set.

If every element in set A is in set B, and there are elements in set A left over it's larger because that's what larger means. {A,B} There are countable and uncountable infinite set's. All countable set's have a bijection with N. However, there are more than two sizes of infinite sets. Real numbers < Imaginary numbers.

Re: On the (Small) Number of Atoms in the Universe

#94

Post by wutbrodo »

jimm wrote: post #87
levemi wrote: post #85

Earlier quoted context omitted.

Within a segment of numbers I understand how there are more rational numbers than integers, but I don't understand it in the context of infinity. How can there be more rational numbers than integers when in both cases there are infinite amounts? Are there mathematical operations or concepts that depend on this (in the context of infinity, not subsets)?

There are different kinds of infinity. The integers are countable, the real numbers aren't. You can prove that if you try to map each integer to some rational number, there will be some rational numbers that are not on that list --- there are more of them. See https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument .

> You can prove that if you try to map each integer to some rational number, there will be some rational numbers that are not on that list --- there are more of them.

Did you mean reals here? There _is_ a (bijection) mapping between integers and rationals.

https://en.wikipedia.org/wiki/Cantor_pairing_function#Cantor...

Re: On the (Small) Number of Atoms in the Universe

#95

Post by dack »

NegativeK wrote: post #14
amelius wrote: post #12

Earlier quoted context omitted.

Related only to large numbers, but is there some theory about generalizing and extending our usual mathematical operators +, ×, and ^ (power)? + applied N times becomes ×N × applied N times becomes ^N ^ applied N times becomes ...? etcetera And would such a theory have any practical use?

Knuth's up-arrow notation is one of a few ways to address the addition->multiplication->exponentiation->tetration->... extensions: https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation As for practicalities, the mathematician in me will let the scientists deal with that.

I like the description of graham's number (by graham) using the up-arrow notation https://youtu.be/GuigptwlVHo?t=31

Re: On the (Small) Number of Atoms in the Universe

#96

Post by pcmonk »

Retric wrote: post #93
pcmonk wrote: post #86

Earlier quoted context omitted.

Where in math is the subset partial ordering used to describe one set as larger than another? Diagonalization isn't showing that a number in set A isn't in set B - that's obviously true for reals and integers, but it's also true for rationals and integers. It's showing that there does not exist a mapping from B to A where there's an element in B for each element in A. We're obviously not using the same definition of…

Cardinality is the number of elements in a set. If every element in set A is in set B, and there are elements in set A left over it's larger because that's what larger means. {A,B} There are countable and uncountable infinite set's. All countable set's have a bijection with N. However, there are more than two sizes of infinite sets. Real numbers < Imaginary numbers.

That's not what larger means, that's what (maps into a) strict subset means. In finite numbers, that's the same as larger (greater cardinality), but it's very much not the case for infinite numbers.

There are more than two sizes of infinite sets, but there are just as many real numbers as imaginary numbers for the same reason there's just as many integers as rational numbers.

Try reading this: https://en.wikipedia.org/wiki/Cardinality#Infinite_sets

Re: On the (Small) Number of Atoms in the Universe

#97

Post by Retric »

pherq wrote: post #90
Retric wrote: post #84

Earlier quoted context omitted.

The set of all prime numbers is contained within the set of rational numbers, but they are rational numbers that are not within the set of prime numbers. Cantor's diagonalization is simply demonstrating that same inequality by showing a number in set A is not in set B. Just because you can map two infinity's to each other does not mean they are of the same size consider: Limit(0->inifinity) of (x - (x/2)) algebraical…

The existence of a bijection between two sets is what "same size" means in set theory. Yes, there are non-prime integers, but you can establish a bijection between the two, so their cardinalities are equal (both have a cardinality of aleph zero). The reals, on the other hand, cannot be placed in a bijection with the natural numbers, and there are therefore "more" reals than naturals (i.e. there is an injection from t…

You're confusing a classification system with size.

Is the set of Real Numbers larger, smaller, or the same size as the set of points in a finite 2d object? Can you setup a bijection in either direction?

Re: On the (Small) Number of Atoms in the Universe

#98

Post by levemi »

ambulancechaser wrote: post #91
levemi wrote: post #85

Earlier quoted context omitted.

Within a segment of numbers I understand how there are more rational numbers than integers, but I don't understand it in the context of infinity. How can there be more rational numbers than integers when in both cases there are infinite amounts? Are there mathematical operations or concepts that depend on this (in the context of infinity, not subsets)?

in mathematics, we lose concept of how many and fall back to cardinality, which has a lose correlation with how many. So asking "are there the same number" of integers as rational numbers gets a little iffy until we make some definitions. We just say that we can create a bijection from integers to rationals. They each index the other, and for each thing in one, there is one and only one thing in the other. Does this…

Thank you for your answer.

> I'm not sure what you mean by mathematical operations or concepts that depend on this.

I suppose I meant to ask if there was any practical application of the concepts you described.

Re: On the (Small) Number of Atoms in the Universe

#99

Post by JBiserkov »

Retric wrote: post #84
pcmonk wrote: post #74

Earlier quoted context omitted.

In what sense are there more rational numbers than prime numbers? They can be put into bijection with each other, so we generally think of them as being same infinity. There are more real nubmers, of course, by Cantor's diagonalization, so your basic point is true.

The set of all prime numbers is contained within the set of rational numbers, but they are rational numbers that are not within the set of prime numbers. Cantor's diagonalization is simply demonstrating that same inequality by showing a number in set A is not in set B. Just because you can map two infinity's to each other does not mean they are of the same size consider: Limit(0->inifinity) of (x - (x/2)) algebraical…

>Cantor's diagonalization is simply demonstrating that same inequality by showing a number in set A is not in set B.

If that were true, why go to all the trouble, just show 1/2 which is not a natural number, or sqrt(2) which is not a rational number.

Cantor's diagonalization is proving that no mapping exists between the natural numbers and the real numbers in [0, 1]; that no matter what mapping you (try to) come up, there will be a number you would miss.

The primes and rationals have the same size (cardinality) as the natural numbers, namely countably infinite. See https://en.wikipedia.org/wiki/Countable_set#Formal_overview_...

Re: On the (Small) Number of Atoms in the Universe

#100

Post by Retric »

pcmonk wrote: post #96
Retric wrote: post #93

Earlier quoted context omitted.

Cardinality is the number of elements in a set. If every element in set A is in set B, and there are elements in set A left over it's larger because that's what larger means. {A,B} There are countable and uncountable infinite set's. All countable set's have a bijection with N. However, there are more than two sizes of infinite sets. Real numbers < Imaginary numbers.

That's not what larger means, that's what (maps into a) strict subset means. In finite numbers, that's the same as larger (greater cardinality), but it's very much not the case for infinite numbers. There are more than two sizes of infinite sets, but there are just as many real numbers as imaginary numbers for the same reason there's just as many integers as rational numbers. Try reading this: https://en.wikipedia.or…

We agree that {A,B,C} has a lower cardinality than {A,B}.

Now, feel free to try and map the set of Real numbers to the set of irrational numbers. ex: e + ei.

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